2024/09/12 by Heiko Gimperlein, Michael Grinfeld, Gimperlein, Heiko +5
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #Elasticity and Wave Propagation #FOS: Mathematics
paper · doi:10.48550/arxiv.2409.07954
openalex publication_date 2024/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A vector field similar to those separately introduced by Artstein and Dafermos is constructed from the tangent to a monotone increasing one-parameter family of non-concentric circles that touch at the common point of intersection taken as the origin. The circles define and space-fill a lens shaped region Ω whose outer and inner boundaries are the greatest and least circles. The double cusp at the origin creates a geometric singularity at which the vector field is indeterminate and has non-unique limiting behaviour. A semi-inverse method that involves the Airy stress function then shows that the vector field corresponds to the displacement vector field for a linear plane compressible non-homogeneous isotropic elastostatic equilibrium problem in Ω whose boundaries are rigidly rotated relative to each other, possibly causing rupture or tearing at the origin. A sequence of solutions is found for which not only are the Lamé parameters strongly-elliptic, but the non-unique limiting behaviour of the displacement is preserved. Other properties of the vector field are also established.